Lemma REF  focuses the study of the leading eigenvalues of \(\widetilde{S}_n\) —those resulting from the signal component of \(\widetilde{X}_n\) —away from the increasing-dimensional sequence \(\widetilde{S}_n\)  to the fixed-dimensional sequence \(\widetilde{M}_n\)  and its associates, \(M_n\)  and \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_n\) . The roots of \(\det \widetilde{M}_n(z)\)  will be shown to localize near the roots of \(\det \hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_n(z)\) . Using the closed form (REF ) of the Stieltjes transform \(\bar{s}_n\) , \(\det \hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_n\)  has explicitly known roots. As [1]} shows, these roots are precisely \(\bar{\lambda }_1, \ldots , \bar{\lambda }_{i_0}\) , introduced in Section REF :
